Polynomial Functions and Their Graphs

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Polynomial Functions and Their Graphs
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POLYNOMIAL FUNCTIONS
A POLYNOMIAL is a monomial or a sum of monomials.
A POLYNOMIAL IN ONE VARIABLE is a polynomial that
contains only one variable.
Example 5x2 3x - 7
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POLYNOMIAL FUNCTIONS
The DEGREE of a polynomial in one variable is the
greatest exponent of its variable.
A LEADING COEFFICIENT is the coefficient of the
term with the highest degree.
What is the degree and leading coefficient of
3x5 3x 2 ?
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POLYNOMIAL FUNCTIONS
A polynomial equation used to represent a
function is called a POLYNOMIAL FUNCTION.
Polynomial functions with a degree of 1 are
called LINEAR POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 2 are
called QUADRATIC POLYNOMIAL FUNCTIONS
Polynomial functions with a degree of 3 are
called CUBIC POLYNOMIAL FUNCTIONS
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POLYNOMIAL FUNCTIONS
A polynomial function is an equation of the form
f (x) anxn an-1xn-1 a2x2 a1x a0
Where the coefficients an, an-1, a1, a0 is
not zero, and the exponents are non-negative
integers.
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Smooth, Continuous Graphs
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POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) 3
Constant Function
Degree 0
Max. Zeros 0
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POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) x 2
Linear Function
Degree 1
Max. Zeros 1
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POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) x2 3x 2
Quadratic Function
Degree 2
Max. Zeros 2
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POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) x3 4x2 2
Cubic Function
Degree 3
Max. Zeros 3
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POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) x4 4x3 2x 1
Quartic Function
Degree 4
Max. Zeros 4
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POLYNOMIAL FUNCTIONS
GENERAL SHAPES OF POLYNOMIAL FUNCTIONS
f(x) x5 4x4 2x3 4x2 x 1
Quintic Function
Degree 5
Max. Zeros 5
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POLYNOMIAL FUNCTIONS
END BEHAVIOR
f(x) x2
Degree Even
Leading Coefficient
End Behavior
As x ? -8 f(x) ? 8
As x ? 8 f(x) ? 8
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POLYNOMIAL FUNCTIONS
END BEHAVIOR
f(x) -x2
Degree Even
Leading Coefficient
End Behavior
As x ? -8 f(x) ? -8
As x ? 8 f(x) ? -8
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POLYNOMIAL FUNCTIONS
END BEHAVIOR
f(x) x3
Degree Odd
Leading Coefficient
End Behavior
As x ? -8 f(x) ? -8
As x ? 8 f(x) ? 8
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POLYNOMIAL FUNCTIONS
END BEHAVIOR
f(x) -x3
Degree Odd
Leading Coefficient
End Behavior
As x ? -8 f(x) ? 8
As x ? 8 f(x) ? -8
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The Leading Coefficient Test
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The Leading Coefficient Test
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Example
Use the Leading Coefficient Test to determine the
end behavior of the graph of the cubic function
f (x) x3 3x2 - x - 3.
Solution Because the degree is odd (n 3)
and the leading coefficient, 1, is positive, the
graph falls to the left and rises to the right,
as shown in the figure.
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Identifying Polynomial Function
Determine whether each function is a polynomial
function. If it is determine its degree.
No. Variable is the exponent.
Yes. Degree is 4.
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Identifying Polynomial Function
No. Exponent has a negative integer.
Yes. Degree is 3.
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Identifying Polynomial Function
No. The term 1/x can not be written in the form x
to the power of n.
No. Same as above.
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Graphing a Polynomial Function Example
Find all zeros of f (x) -x4 4x3 - 4x2.
Solution We find the zeros of f by setting
f (x) equal to 0.
-x4 4x3 - 4x2 0 We now have a
polynomial equation.
x4 - 4x3 4x2 0 Multiply both
sides by -1. (optional step)
x2(x2 - 4x 4) 0 Factor out x2.
x2(x - 2)2 0 Factor completely.
x2 0 or (x - 2)2 0 Set each
factor equal to zero.
x 0 x 2 Solve for
x.
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Example cont.
Graph f (x) x4 - 2x2 1.
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Example cont.
Graph f (x) x4 - 2x2 1.
Solution
Step 2 Find the x-intercepts (zeros of the
function) by setting f (x) 0.
x4 - 2x2 1 0
(x2 - 1)(x2 - 1) 0 Factor.
(x 1)(x - 1)(x 1)(x - 1) 0 Factor
completely.
(x 1)2(x - 1)2 0 Express the
factoring in more compact notation.
(x 1)2 0 or (x - 1)2 0 Set
each factor equal to zero.
x -1 x 1 Solve
for x.
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Example cont.
Graph f (x) x4 - 2x2 1.
Solution
Step 2 We see that -1 and 1 are both repeated
zeros with multiplicity 2. Because of the even
multiplicity, the graph touches the x-axis at -1
and 1 and turns around. Furthermore, the graph
tends to flatten out at these zeros with
multiplicity greater than one
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Example cont.
Graph f (x) x4 - 2x2 1.
Solution
Step 3 Find the y-intercept. Replace x with
0 in f (x) -x 4x - 1.
f (0) 04 - 2 02 1 1
There is a y-intercept at 1, so the graph passes
through (0, 1).
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Example cont.
Graph f (x) x4 - 2x2 1.
Solution